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Grouped Variable Selection with Discrete Optimization: Computational and Statistical Perspectives

14 Apr 2021arXiv:2104.07084archive 2025-07-28

Hussein Hazimeh, Rahul Mazumder, Peter Radchenko

We present a new algorithmic framework for grouped variable selection that is based on discrete mathematical optimization. While there exist several appealing approaches based on convex relaxations and nonconvex heuristics, we focus on optimal solutions for the ℓ₀-regularized formulation, a problem that is relatively unexplored due to computational challenges. Our methodology covers both high-dimensional linear regression and nonparametric sparse additive modeling with smooth components. Our algorithmic framework consists of approximate and exact algorithms. The approximate algorithms are based on coordinate descent and local search, with runtimes comparable to popular sparse learning algorithms. Our exact algorithm is based on a standalone branch-and-bound (BnB) framework, which can solve the associated mixed integer programming (MIP) problem to certified optimality. By exploiting the problem structure, our custom BnB algorithm can solve to optimality problem instances with 5 ×10⁶ features and 10³ observations in minutes to hours -- over $1000$ times larger than what is currently possible using state-of-the-art commercial MIP solvers. We also explore statistical properties of the ℓ₀-based estimators. We demonstrate, theoretically and empirically, that our proposed estimators have an edge over popular group-sparse estimators in terms of statistical performance in various regimes. We provide an open-source implementation of our proposed framework.

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